\[(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
Test:
simple fma test
Bits:
128 bits
Bits error versus x
Bits error versus y
Bits error versus z
Time: 8.0 s
Input Error: 61.7
Output Error: 23.6
Log: ⚲
Profile: 🕒
\((\left(\frac{-1}{x}\right) * \left(-\frac{1}{y}\right) + \left(-\frac{1}{z}\right))_* - \left(\left(1 + \frac{1}{z}\right) + \frac{1}{x \cdot y}\right)\)
  1. Started with
    \[(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right)\]
    61.7
  2. Applied taylor to get
    \[(x * y + z)_* - \left(1 + \left(x \cdot y + z\right)\right) \leadsto (\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - \left(y \cdot x + \left(1 + z\right)\right)\]
    60.1
  3. Taylor expanded around -inf to get
    \[\color{red}{(\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - \left(y \cdot x + \left(1 + z\right)\right)} \leadsto \color{blue}{(\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - \left(y \cdot x + \left(1 + z\right)\right)}\]
    60.1
  4. Applied taylor to get
    \[(\left(\frac{-1}{x}\right) * \left(\frac{-1}{y}\right) + \left(\frac{-1}{z}\right))_* - \left(y \cdot x + \left(1 + z\right)\right) \leadsto (\left(-1 \cdot x\right) * \left(-1 \cdot y\right) + \left(-1 \cdot z\right))_* - \left(y \cdot x + \left(1 + z\right)\right)\]
    62.2
  5. Taylor expanded around inf to get
    \[\color{red}{(\left(-1 \cdot x\right) * \left(-1 \cdot y\right) + \left(-1 \cdot z\right))_*} - \left(y \cdot x + \left(1 + z\right)\right) \leadsto \color{blue}{(\left(-1 \cdot x\right) * \left(-1 \cdot y\right) + \left(-1 \cdot z\right))_*} - \left(y \cdot x + \left(1 + z\right)\right)\]
    62.2
  6. Applied simplify to get
    \[\color{red}{(\left(-1 \cdot x\right) * \left(-1 \cdot y\right) + \left(-1 \cdot z\right))_* - \left(y \cdot x + \left(1 + z\right)\right)} \leadsto \color{blue}{(\left(-x\right) * \left(-y\right) + \left(-z\right))_* - \left(\left(z + 1\right) + y \cdot x\right)}\]
    62.2
  7. Using strategy rm
    62.2
  8. Applied add-cube-cbrt to get
    \[\color{red}{(\left(-x\right) * \left(-y\right) + \left(-z\right))_* - \left(\left(z + 1\right) + y \cdot x\right)} \leadsto \color{blue}{{\left(\sqrt[3]{(\left(-x\right) * \left(-y\right) + \left(-z\right))_* - \left(\left(z + 1\right) + y \cdot x\right)}\right)}^3}\]
    62.2
  9. Applied taylor to get
    \[{\left(\sqrt[3]{(\left(-x\right) * \left(-y\right) + \left(-z\right))_* - \left(\left(z + 1\right) + y \cdot x\right)}\right)}^3 \leadsto {\left(\sqrt[3]{(\left(-\frac{1}{x}\right) * \left(-\frac{1}{y}\right) + \left(-\frac{1}{z}\right))_* - \left(\frac{1}{z} + \left(\frac{1}{y \cdot x} + 1\right)\right)}\right)}^3\]
    24.1
  10. Taylor expanded around inf to get
    \[{\color{red}{\left(\sqrt[3]{(\left(-\frac{1}{x}\right) * \left(-\frac{1}{y}\right) + \left(-\frac{1}{z}\right))_* - \left(\frac{1}{z} + \left(\frac{1}{y \cdot x} + 1\right)\right)}\right)}}^3 \leadsto {\color{blue}{\left(\sqrt[3]{(\left(-\frac{1}{x}\right) * \left(-\frac{1}{y}\right) + \left(-\frac{1}{z}\right))_* - \left(\frac{1}{z} + \left(\frac{1}{y \cdot x} + 1\right)\right)}\right)}}^3\]
    24.1
  11. Applied simplify to get
    \[{\left(\sqrt[3]{(\left(-\frac{1}{x}\right) * \left(-\frac{1}{y}\right) + \left(-\frac{1}{z}\right))_* - \left(\frac{1}{z} + \left(\frac{1}{y \cdot x} + 1\right)\right)}\right)}^3 \leadsto (\left(\frac{-1}{x}\right) * \left(-\frac{1}{y}\right) + \left(-\frac{1}{z}\right))_* - \left(\left(1 + \frac{1}{z}\right) + \frac{1}{x \cdot y}\right)\]
    23.6
  12. Applied final simplification

Original test:


(lambda ((x default) (y default) (z default))
  #:name "simple fma test"
  (- (fma x y z) (+ 1 (+ (* x y) z)))
  #:target
  -1)